## LCM of Algebraic Expressions

To calculate the LCM (Lowest Common Multiple) of the given algebraic expressions, we should convert the algebraic expressions into their simplest factors. And, then the product of their common factors and the remaining factors will give the LCM of the given algebraic expressions.

LCM = Common factors × Remaining factors

For example: x2 + 5x + 6 and x2 + 3x + 2 are two algebraic expressions. x2 + 5x + 6 = (x + 2)(x + 3) and x2 + 3x + 2 = (x + 1)(x+2) are their simplest factors.  And, (x + 1)(x + 2)(x + 3) is the product of their common and remaining factors. So, LCM = (x + 1)(x + 2)(x + 3).

While finding LCM, we should use the following steps:

Steps:

1.    Factorize the given algebraic expressions.

2.    Take out the common factors and then the remaining factors.

3.    The product of common and remaining factors will give the required LCM.

The concept of LCM will be more clear from the following worked out examples.

### Workout Examples

Example 1: Find the LCM of 3x2yz, 4y2z and 5xz2

Solution: Here,

1st expression = 3x2yz = 3 × x × x × y × z

2nd expression = 4y2z = 2 × 2 × y × y × z

3rd expression = 5xz2 = 5 × x × z × z

LCM = 3 × 2 × 2 × 5 × x × x × y × y × z × z

= 60x2y2z2

Example 2: Find the LCM of m4 – 4m2 and 3m2 + 6m

Solution: Here,

1st expression = m4 – 4m2

= m2(m2 – 4)

= m2(m2 – 22)

= m × m(m + 2)(m – 2)

2nd expression = 3m2 + 6m

= 3m(m + 2)

LCM = 3m × m(m + 2)(m – 2)

= 3m2(m + 2)(m – 2)

Example 3: Find the LCM of a2 – 3a + 2, a4 + a3 – 6a2 and a3 + 2a2 – 3a

Solution: Here,

1st expression = a2 – 3a + 2

= a2 – (2 + 1)a + 2

= a2 – 2a – a + 2

= a(a – 2) – 1(a – 2)

= (a – 2)(a – 1)

2nd expression = a4 + a3 – 6a2

= a2(a2 + a – 6)

= a2{a2 + (3 – 2)a – 6}

= a2(a2 + 3a – 2a – 6)

= a2{a(a + 3) – 2(a + 3)}

= a × a(a + 3)(a – 2)

3rd expression = a3 + 2a2 – 3a

= a(a2 + 2a – 3)

= a{a2 + (3 – 1)a – 3}

= a(a2 + 3a – a – 3)

= a{a(a + 3) – 1(a + 3)}

= a(a + 3)(a – 1)

LCM = a × a(a – 1)(a – 2)(a + 3)

= a2(a – 1)(a – 2)(a + 3)

Example 4: Find the LCM of x2 – 5x + 6, x2 + 4x – 12 and x2 – 2x

Solution: Here,

1st expression = x2 – 5x + 6

= x2 – (3 + 2)x + 6

= x2 – 3x – 2x + 6

= x(x – 3) – 2(x – 3)

= (x – 3)(x – 2)

2nd expression = x2 + 4x – 12

= x2 + (6 – 2)x – 12

= x2 + 6x – 2x - 12

= x(x + 6) – 2(x + 6)

= (x + 6)(x – 2)

3rd expression = x2 – 2x

= x(x – 2)

LCM = x(x – 2)(x – 3)(x + 6)

Example 5: Find the LCM of a2 – b2 – 2bc – c2, b2 – c2 – 2ca – a2 and c2 – a2 – 2ab – b2

Solution: Here,

1st expression = a2 – b2 – 2bc – c2

= a2 – (b2 + 2bc + c2)

= a2 – (b + c)2

= (a+b+c)(a–b–c)

2nd expression = b2 – c2 – 2ca – a2

= b2 – (c2 + 2ca + a2)

= b2 – (c + a)2

= (b+c+a)(b–c–a)

= (a+b+c)(b–c–a)

3rd expression = c2 – a2 – 2ab – b2

= c2 – (a2 + 2ab + b2)

= c2 – (a + b)2

= (c+a+b)(c–a–b)

= (a+b+c)(c–a–b)

LCM = (a + b + c)(a – b – c)(b – c – a)(c – a – b)

Example 6: Find the LCM of a3 – 2a2b + 2ab2 – b3, a4 + b4 + a2b2, 4a4b + 4ab4

Solution: Here,

1st expression = a3 – 2a2b + 2ab2 – b3

= a3 – b3 – 2a2b + 2ab2

= (a–b)(a2+ab+b2) – 2ab(a–b)

= (a–b)(a2+ab+b2–2ab)

= (a – b)(a2 – ab + b2)

2nd expression = a4 + b4 + a2b2

= (a2)2 + 2a2b2 + (b2)2 – a2b2

= (a2 + b2)2 –(ab)2

= (a2+b2+ab)(a2+b2–ab)

= (a2+ab+b2)(a2–ab+b2)

3rd expression = 4a4b + 4ab4

= 4ab(a3 + b3)

= 4ab(a + b)(a2 – ab + b2)

LCM = 4ab(a + b)(a – b)(a2 + ab + b2)(a2 – ab + b2)

Example 7: Find the LCM of a2 – 18a – 19 + 20b – b2, a2 + a – b2 + b, 4a2 – 4b2 + 8b - 4

Solution: Here,

1st expression = a2 – 2.a.9 + 92 – 100 + 20b – b2

= (a – 9)2 – (102 – 2.10.b + b2)

= (a – 9)2 – (10 – b)2

= {(a – 9) + (10 – b)}{(a – 9) – (10 – b)}

= (a – 9 + 10 – b)(a – 9 – 10 + b)

= (a – b + 1)(a + b – 19)

2nd expression = a2 + a – b2 + b

= a2 – b2 + a + b

= (a+b)(a–b) + 1(a+b)

= (a + b)(a – b + 1)

3rd expression = 4a2 – 4b2 + 8b - 4

= 4(a2 – b2 + 2b – 1)

= 4{a2 – (b2 – 2b + 1)}

= 4{a2 – (b – 1)2}

= 4(a + b – 1)(a – b + 1)

LCM = 4(a + b)(a + b – 1)(a – b + 1)(a + b – 19)

Example 8: Find the LCM of 1 + 4a + 4a2 – 16a4, 1 + 2a – 8a3 – 16a4 and 1 – 8a3

Solution: Here,

1st expression = 1 + 4a + 4a2 – 16a4

= 12 + 2.1.2a + (2a)2 – (4a2)2

= (1 + 2a)2 – (4a2)2

= (1 + 2a + 4a2)(1 + 2a – 4a2)

2nd expression = 1 + 2a – 8a3 – 16a4

= 1(1 + 2a) – 8a3(1 + 2a)

= (1 + 2a)(1 – 8a3)

= (1 + 2a){13 – (2a)3}

=(1 + 2a)(1 – 2a)(1 + 2a + 4a2)

3rd expression = 1 – 8a3

= 13 – (2a)3

= (1 – 2a)(1 + 2a + 4a2)

LCM = (1 + 2a)(1 – 2a)(1 + 2a + 4a2)(1 + 2a – 4a2)

You can comment your questions or problems regarding the LCM of algebraic expressions here.